!"#$%& IIT Kanpur. !"#$%&. Kanpur, How spins become pairs: Composite and magnetic pairing in the 115 Heavy Fermion Superconductors

How spins become pairs: Composite and magnetic pairing in the 115 Heavy Fermion Superconductors!"#$%& IIT Kanpur Feb 6 2010 Interaction, Instability and Transport!"#$%&. Kanpur, 1857.

How spins become pairs: Composite and magnetic pairing in the 115 Heavy Fermion Superconductors!"#$%& IIT Kanpur Feb 6 2010 Interaction, Instability and Transport M. Dzero R. Flint P. Coleman Local Moment Center for Materials Theory 4.5K Heavy Fermion S.C NpAl2Pd5

Acknowledgments: Nature Physics 4, 643 (2008). arxiv:0912.2339 Rebecca Flint Maxim Dzero (U. Maryland).

Acknowledgments: Nature Physics 4, 643 (2008). arxiv:0912.2339 Rebecca Flint Maxim Dzero (U. Maryland). Scott Thomas (Rutgers) Pascoal Pagliuso & group (Unicamp) Joe Thompson (LANL) Filip Ronning (LANL) E. D. Bauer (LANL) Zachary Fisk (UCIrvine) Cigdem Capan (UCIrvine) Alexei Tsvelik (BNL) Natan Andrei (Rutgers) Hae Young Kee (Toronto) PRB 60, 3609, (1998).

Outline

Outline! The materials: Why the 115s are so special?

Outline! The materials: Why the 115s are so special?! The concept: How do spins pair?! Composite pairing and the Two Channel Kondo model

Outline! The materials: Why the 115s are so special?! The concept: How do spins pair?! Composite pairing and the Two Channel Kondo model! The tool: Symplectic-N

Outline! The materials: Why the 115s are so special?! The concept: How do spins pair?! Composite pairing and the Two Channel Kondo model! The tool: Symplectic-N! The results: Composite pairing phase diagram

Conventional Heavy Fermion Superconductivity Example: UPt 3 T* ~ 100K, T C =.56K Pauli Susceptibility Curie

Conventional Heavy Fermion Superconductivity Example: UPt 3 Stage one: QP formation T* ~ 100K, T C =.56K Pauli paramagnetism fully Pauli Susceptibility developed by 30K~50 T C Curie

Conventional Heavy Fermion Superconductivity Example: UPt 3 Stage one: QP formation T* ~ 100K, T C =.56K Pauli paramagnetism fully Pauli Susceptibility developed by 30K~50 T C Curie Spin Fluctuations

Conventional Heavy Fermion Superconductivity Example: UPt 3 Stage one: QP formation Pauli T* ~ 100K, T C =.56K Susceptibility Curie Pauli paramagnetism fully developed by 30K~50 T C Stage two Unconventional superconductivity mediated by spin fluctuations

Conventional Heavy Fermion Superconductivity Example: UPt 3 T* ~ 100K, T C =.56K Stage one: QP formation Pauli paramagnetism fully developed by 30K~50 T C Stage two Unconventional superconductivity mediated by spin fluctuations Led to proposal that AFM spin fluctuations drive d-wave pairing Nodal SC Beal-Monod, Bourbonnais and Emery Scalapino, Loh and Hirsch Miyake, Schmitt-Rink, and Varma 1986 Kohori 1988

115 materials: a new family of superconductors In

115 materials: a new family of superconductors Oxides T c? YBa 2 Cu 3 O 7 92K In Ba2201 12K

115 materials: a new family of superconductors Oxides 115 Intermetallics T c T c? YBa 2 Cu 3 O 7 92K Mathur et al, Nature 394, 39 (1998) In Ba2201 12K 0.2K CeIn 3 96

115 materials: a new family of superconductors Ce Co Oxides 115 Intermetallics T c T c? YBa 2 Cu 3 O 7 92K 2K CeCoIn 5 01 In Ba2201 12K 0.2K CeIn 3 96

115 materials: a new family of superconductors Ce Co Oxides 115 Intermetallics Tuson Park, (2007). CeRhIn 5 T c T c? YBa 2 Cu 3 O 7 92K 2K CeCoIn 5 01 In Ba2201 12K 0.2K CeIn 3 96

115 materials: a new family of superconductors Oxides 115 Intermetallics T c T c? 18.5K PuCoGa 5 02 YBa 2 Cu 3 O 7 92K 2K CeCoIn 5 01 In Ba2201 12K 0.2K CeIn 3 96

115 materials: a new family of superconductors Oxides 115 Intermetallics T c T c? 18.5K PuCoGa 5 02 NpAl 2 Pd 5 4.5K 07 YBa 2 Cu 3 O 7 92K 2K CeCoIn 5 01 4.5K Heavy Fermion S.C NpAl2Pd5 In D. Aoki et al., JPSJ 76 (2007) Ba2201 12K 0.2K CeIn 3 96

The 115s: local moments at T C NpPd 5 Al 2 T C = 4.5K Curie No Pauli paramagnetism ~1/3 R ln(2) Large condensation entropy Aoki et al 2007

The 115s: local moments at T C NpPd 5 Al 2 T C = 4.5K CeCoIn 5 T C = 2.3K Ce Co In ~1/3 R ln(2) ~1/3 R ln(2) Aoki et al 2007 Petrovic et al 2001

The 115s: local moments at T C Ce In NpPd 5 Al 2 T C = 4.5K CeCoIn 5 T C = 2.3K Co Follow spins below T C ~1/3 R ln(2) 1/3 R ln(2) 1/3 R ln(2) Knight Shift Aoki et al 2007 Petrovic et al 2001 Aoki et al 2007 Curro et al 2001

Curro et al 2001 The 115s: local moments at T C Above T C ~1/3 R ln(2) Curie Free local moments down to T C Aoki et al 2007 Below T C Knight Shift Spin singlet, nodal SC Spins pair and quench simultaneously Curro et al 2001 Sarrao and Thompson 2007 How are spins incorporated into the pair?

Kondo effect and Heavy Fermions Review: cond-mat/0612006 Nozieres 74

Kondo effect and Heavy Fermions Review: cond-mat/0612006 H = Jσ (0) S J. Kondo 64 Nozieres 74

Kondo effect and Heavy Fermions Review: cond-mat/0612006 H = Jσ (0) S T K F e F J J. Kondo 64 Nozieres 74

Kondo effect and Heavy Fermions Review: cond-mat/0612006 χ T K 1 T K H = Jσ (0) S χ 1/T T K F e F J T K J. Kondo 64 T Nozieres 74

Kondo effect and Heavy Fermions Review: cond-mat/0612006

Kondo effect and Heavy Fermions Review: cond-mat/0612006 PuCoGa 5 H = Jσ (0) S

Kondo effect and Heavy Fermions Review: cond-mat/0612006 PuCoGa 5 H = Jσ (0) S Directly Curie PM to SC

Kondo effect and Heavy Fermions Review: cond-mat/0612006 PuCoGa 5 H = Jσ (0) S S.C. Directly Curie PM to SC

Kondo effect and Heavy Fermions Review: cond-mat/0612006 PuCoGa 5 H = Jσ (0) S S.C. How do spins quench and form pairs?

Outline! The materials: Why the 115s are so special?! The concept: How do spins pair?! Composite pairing and the Two Channel Kondo model model! The tool: Symplectic-N! The results: Composite pairing phase diagram! Hybridization: of magnetic and composite pairing T C - - - -

Cotunneling and composite fermions

Cotunneling and composite fermions van der Wiel et al, Science (2000)

Cotunneling and composite fermions van der Wiel et al, Science (2000) + T>>TK: Coulomb Blockade

Cotunneling and composite fermions van der Wiel et al, Science (2000) T<<TK

Cotunneling and composite fermions van der Wiel et al, Science (2000) Cotunneling (Glazman +Pustilnik) (quantum dots) f = c S T<<TK

Cotunneling and composite fermions Cotunneling Cotunneling (Glazman +Pustilnik) (quantum dots) Heavy electron = (electron x spinflip) f = c S

Cotunneling and copairing: Copairing Cotunneling (Glazman +Pustilnik) (quantum dots) Heavy Cooper pair = (pair x spinflip) f = c S

Cotunneling and copairing: Copairing Cotunneling (Glazman +Pustilnik) (quantum dots) Heavy Cooper pair = (pair x spinflip) f = c S Ψ = c 1 c 2 S +

Cotunneling and copairing: Copairing Cotunneling (Glazman +Pustilnik) (quantum dots) Heavy Cooper pair = (pair x spinflip) f = c S Ψ = c 1 c 2 S + Abrahams, Balatsky, Scalapino, Schrieffer 1995

Two-channel model for composite pairing.

The Two Channel Kondo Model Wannier functions at site j:

The Two Channel Kondo Model Wannier functions at site j:! Both {

H = k k c kσ c kσ + 1 N cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

Impurity: quantum critical point for J 1 = J 2 Nozieres and Blandin 1980 T/TK FL1 FL2 J2/J1 H = k k c kσ c kσ + 1 N cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

Impurity: quantum critical point for J 1 = J 2 Nozieres and Blandin 1980 Singular composite pair fluctuations Emery and Kivelson 1992 T/TK FL1 FL2 J2/J1 H = k k c kσ c kσ + 1 N cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

Impurity: quantum critical point for J 1 = J 2 Nozieres and Blandin 1980 Singular composite pair fluctuations Emery and Kivelson 1992 T/TK Avoided in the lattice by composite pairing. FL1 FL2 J2/J1 H = k k c kσ c kσ + 1 N cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

Impurity: quantum critical point for J 1 = J 2 Nozieres and Blandin 1980 Singular composite pair fluctuations Emery and Kivelson 1992 T/TK Avoided in the lattice by composite pairing. FL1 - - FL2 J2/J1 H = k k c kσ c kσ + 1 N cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

H = k cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k c kσ c kσ + 1 N k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

S[ψ] So how can we solve this τ,x model? ψ(x, τ) Wild quantum fluctuations! H = k cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k c kσ c kσ + 1 N k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

Method: symplectic large N Flint & Coleman ʼ08 PRB 79, 014424(2009) S[ψ] So how can we solve this τ,x model? ψ(x, τ) H = k cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k c kσ c kσ + 1 N k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

Method: symplectic large N Flint & Coleman ʼ08 PRB 79, 014424(2009) S[ψ] τ,x Large N limit ψ(x, τ) H = k cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k c kσ c kσ + 1 N k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

S[ψ] τ,x Large N limit 1 N eff ψ(x, τ) H = k cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k c kσ c kσ + 1 N k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

S[ψ] τ,x Large N limit 1 N eff classical path ψ(x, τ) H = k cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k c kσ c kσ + 1 N k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk S ba (j) c k e ik x j

Symplectic N (large N + time reversal symmetry) R. Flint and PC 08 S[ψ] τ,x Large N limit 1 N eff classical path ψ(x, τ) H = k cf Cox, Pang, Jarell (96) PC, Kee, Andrei, Tsvelik (98) k c kσ c kσ + 1 N k,k J 1 ψ 1a (j)ψ 1b(j) + J 2 ψ 2a (j)ψ 2b(j) Single FS, two channels. ψ Γ (j) = 1 V k γ Γk? S ba (j) c k e ik x j

The one channel model in Symplectic N H I = J K N = J K N ψ a ψ b S ba f b f a ã bf af b (ψ f)(f ψ) + (ψ σ 2 f )(fσ 2 ψ).

The one channel model in Symplectic N H I = J K N = J K N ψ a ψ b S ba f b f a ã bf af b (ψ f)(f ψ) + (ψ σ 2 f )(fσ 2 ψ) Hybridization.

The one channel model in Symplectic N H I = J K N = J K N ψ a ψ b S ba f b f a ã bf af b (ψ f)(f ψ) + (ψ σ 2 f )(fσ 2 ψ). Hybridization Pairing

The one channel model in Symplectic N H I = J K N = J K N ψ a ψ b S ba f b f a ã bf af b (ψ f)(f ψ) + (ψ σ 2 f )(fσ 2 ψ). Hybridization Pairing H I [V f + ( fσ 2 )]ψ + H.c + N V V + J K.

The one channel model in Symplectic N H I = J K N = J K N ψ a ψ b S ba f b f a ã bf af b (ψ f)(f ψ) + (ψ σ 2 f )(fσ 2 ψ). Hybridization Pairing Aside: Continuous particle-hole symmetry Reflects the neutrality of the spin Pairing can be gauged away N= 2: Affleck, Zou, Hsu and Anderson 1988 H I [V f + ( fσ 2 )]ψ + H.c + N V V + J K.

The one channel model in Symplectic N H I = J K N = J K N ψ a ψ b S ba f b f a ã bf af b (ψ f)(f ψ) + (ψ σ 2 f )(fσ 2 ψ). Hybridization Pairing Aside: Continuous particle-hole symmetry N= 2: Affleck, Zou, Hsu and Anderson 1988 Reflects the neutrality of the spin Pairing can be gauged away -recovering results of SU(N) large N.

The 2-channel model:

The 2-channel model: H I + [V 1 f +( 1 fσ 2 )]ψ 1 +H.c [V 2 f +( 2 fσ 2 )]ψ 2 +H.c + N + N V1 V 1 + 1 1 J 1 V2 V 2 + 2 2 J 2

The 2-channel model: H I + [V 1 f +( 1 fσ 2 )]ψ 1 +H.c [V 2 f +( 2 fσ 2 )]ψ 2 +H.c + N + N V1 V 1 + 1 1 J 1 V2 V 2 + 2 2 J 2 Gauge fixing does not remove the pairing [V 1 f ψ 1 +( 2 fσ 2 )ψ 2 ]+H.c + N V1 V 1 + 2 2 J 1 J 2

The 2-channel model: H I + [V 1 f +( 1 fσ 2 )]ψ 1 +H.c [V 2 f +( 2 fσ 2 )]ψ 2 +H.c + N + N V1 V 1 + 1 1 J 1 V2 V 2 + 2 2 J 2 Gauge fixing does not remove the pairing [V 1 f ψ 1 +( 2 fσ 2 )ψ 2 ]+H.c + N V1 V 1 + 2 2 J 1 J 2 The gauge-invariant cross term formation of composite pairs V 1 2 V 2 1 describe the overscreened S ψ 1 σ 2 σ ψ 2 V 1 2 V 2 1 PC, Kee, Andrei & Tsvelik. (97)

The Phase Diagram 1 V 1 " 0 # 2 " 0 V 1! 2 " 0 0 0.2 1 5!

The Phase Diagram 1 Heavy Fermi Liquid One Heavy Fermi Liquid Two V 1 " 0 # 2 " 0 - V 1! 2 " 0 - - - 0 0.2 1 5!

But how to unify magnetically mediated pairing and composite pairing in the Ce 115s?

But how to unify magnetically mediated pairing and composite pairing in the Ce 115s? Magnetically mediated pairing

But how to unify magnetically mediated pairing and composite pairing in the Ce 115s? Magnetically mediated pairing Composite pairing

The Two channel Kondo-Heisenberg Model Two channel Kondo Model Heisenberg Model

The Two channel Kondo-Heisenberg Model

The Two channel Kondo-Heisenberg Model - Spin liquid state

The Two channel Kondo-Heisenberg Model - + + = - - Spin liquid state d-wave Anderson 1973 Baskaran, Zou and Anderson 1987

The Two channel Kondo-Heisenberg Model - + + - Spin liquid state d-wave Anderson 1973 Baskaran, Zou and Anderson 1987

The Two channel Kondo-Heisenberg Model - V 1 - + + - V 1 -

The Two channel Kondo-Heisenberg Model Valence bonds become charged, mobile Cooper pairs + - V 1 - + - V 1 -

The Two channel Kondo-Heisenberg Model Valence bonds become charged, mobile Cooper pairs Inheriting d-wave nature from the antiferromagnetism - V 1 + - - V 1 - +

The Two channel Kondo-Heisenberg Model Valence bonds become charged, mobile Cooper pairs Inheriting d-wave nature from the antiferromagnetism - V 1 + - - V 1 - + Two band version of RVB pairing - - Anderson 1987 Andrei and Coleman 1989 Miyake, Schmitt-Rink, Varma 1986

The Two channel Kondo-Heisenberg Model Spin Liquid # H V 1 # H Heavy Fermi Liquid V 1 Superconductor - - Andrei and Coleman 1989

The Phase Diagram Magnetic pairing V 1 # H " 0

The Phase Diagram Composite pairing V 1 # 2 " 0

The Phase Diagram Decoupled magnetic and composite pairing V 1 # H +V 1 # 2 " 0

The Phase Diagram Hybrid pairing V 1 # 2 # H " 0

The Phase Diagram Composite and d-wave pairing mutually reinforce. Hybrid pairing V 1 # 2 # H " 0

Landau Theory

Comparision to the Ce 115s Sarrao and Thompson 2007

Comparision to the Ce 115s Chemical pressure traces a path through phase space Sarrao and Thompson 2007

Comparision to the Ce 115s Chemical pressure traces a path through phase space CeCoIn 5 CeIrIn 5 CeRhIn 5 Sarrao and Thompson 2007

Comparision to the Ce 115s Chemical pressure traces a path through phase space? Sarrao and Thompson 2007

Comparision to the Ce 115s Chemical pressure traces a path through phase space Sarrao and Thompson 2007

Experimental implications! Valence shift (#n f ) associated with Kondo physics Gunnarsson and Schonhammer 1983

Experimental implications! Valence shift (#n f ) associated with Kondo physics! Sharp valence shift of opposite sign expected at T C! Observable in core level spectroscopy n f 1 X-ray 0 T C T K T Gunnarsson and Schonhammer 1983

Experimental implications! Valence shift (#n f ) associated with Kondo physics! Sharp valence shift of opposite sign expected at T C! Observable in core level spectroscopy 1 X-ray n f! Orbital Knight shift in NQR? 0 T T C T K

Conclusions! Spins quench as they pair in the 115s - -! Must be incorporated directly into the condensate! Two channel physics leads to composite pairs! First controlled mean field theory of heavy fermion superconductivity - Symplectic-N! Composite and magnetic pairing cooperate to enhance T C - - - -

Discussion: Thank You!